Consistency of a needlet spectral estimator on the sphere

The angular power spectrum of a stationary random field on the sphere is estimated from the needlet coefficients of a single realization, observed with increasingly fine resolution. The estimator we consider is similar to the one recently used in practice by (Fa\"{y} et al. 2008) to estimate the power spectrum of the Cosmic Microwave Background. The consistency of the estimator, in the asymptotics of high frequencies, is proved for a model with a stationary Gaussian field corrupted by heteroscedastic noise and missing data.